Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 508 · 40 questions

How to solve it (Patterns repeat): Divisibility: 3 and 9 — digit sum; 4 — last two digits; 8 — last three digits; 6 — even and divisible by 3; 11 — alternate digit sums differ by 0 or a multiple of 11. Remainder: find the biggest multiple not more than the number and take it away; to make a number divisible, add divisor − remainder. Units digit of a power: write the cycle (7, 9, 3, 1 for 7), divide the power by the cycle length and use the remainder.

1. 1.What is the units digit of 3^54? (a) 9 (b) 3 (c) 7 (d) 1
2. 2.What is the units digit of 9^28? (a) 0 (b) 9 (c) 2 (d) 1
3. 3.What is the smallest number you must add to 472 to make it divisible by 6? (a) 2 (b) 4 (c) 3 (d) 1
4. 4.What is the remainder when 184 is divided by 12? (a) 4 (b) 3 (c) 8 (d) 5
5. 5.What is the remainder when 433 is divided by 8? (a) 1 (b) 2 (c) 7 (d) 0
6. 6.What is the smallest number you must add to 233 to make it divisible by 4? (a) 2 (b) 1 (c) 3
7. 7.What is the remainder when 169 is divided by 3? (a) 1 (b) 0 (c) 2
8. 8.What is the remainder when 162 is divided by 10? (a) 1 (b) 8 (c) 3 (d) 2
9. 9.What is the remainder when 69 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
10. 10.What is the smallest number you must add to 179 to make it divisible by 3? (a) 1 (b) 2
11. 11.Which of these numbers is divisible by 6? (a) 2374 (b) 2387 (c) 2376 (d) 2378
12. 12.What is the smallest number you must add to 472 to make it divisible by 7? (a) 3 (b) 5 (c) 6 (d) 4
13. 13.What is the remainder when 194 is divided by 12? (a) 1 (b) 2 (c) 3 (d) 10
14. 14.Which of these numbers is divisible by 8? (a) 807 (b) 794 (c) 786 (d) 792
15. 15.Which of these numbers is divisible by 3? (a) 984 (b) 986 (c) 983 (d) 988
16. 16.What is the remainder when 151 is divided by 9? (a) 6 (b) 7 (c) 2 (d) 8
17. 17.What is the remainder when 172 is divided by 8? (a) 5 (b) 6 (c) 3 (d) 4
18. 18.What is the smallest number you must add to 468 to make it divisible by 8? (a) 3 (b) 5 (c) 6 (d) 4
19. 19.What is the remainder when 139 is divided by 8? (a) 2 (b) 4 (c) 5 (d) 3
20. 20.What is the remainder when 401 is divided by 12? (a) 5 (b) 7 (c) 4 (d) 6
21. 21.Which of these numbers is divisible by 3? (a) 2357 (b) 2360 (c) 2358 (d) 2363
22. 22.What is the smallest number you must add to 57 to make it divisible by 4? (a) 2 (b) 1 (c) 3
23. 23.What is the smallest number you must add to 99 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
24. 24.What is the remainder when 282 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
25. 25.What is the smallest number you must add to 278 to make it divisible by 4? (a) 3 (b) 1 (c) 2
26. 26.What is the remainder when 61 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
27. 27.Which of these numbers is divisible by 3? (a) 1786 (b) 1781 (c) 1782 (d) 1784
28. 28.Which of these numbers is divisible by 8? (a) 2329 (b) 2320 (c) 2316 (d) 2325
29. 29.What is the units digit of 2^55? (a) 8 (b) 6 (c) 4 (d) 2
30. 30.What is the remainder when 439 is divided by 11? (a) 8 (b) 9 (c) 1 (d) 10
31. 31.What is the smallest number you must add to 89 to make it divisible by 3? (a) 2 (b) 1
32. 32.What is the smallest number you must add to 284 to make it divisible by 8? (a) 3 (b) 6 (c) 4 (d) 5
33. 33.What is the units digit of 7^29? (a) 9 (b) 3 (c) 1 (d) 7
34. 34.What is the smallest number you must add to 145 to make it divisible by 11? (a) 9 (b) 10 (c) 2 (d) 8
35. 35.What is the units digit of 3^38? (a) 1 (b) 9 (c) 3 (d) 7
36. 36.What is the units digit of 7^20? (a) 9 (b) 7 (c) 3 (d) 1
37. 37.What is the smallest number you must add to 99 to make it divisible by 7? (a) 1 (b) 4 (c) 6 (d) 5
38. 38.What is the remainder when 211 is divided by 11? (a) 9 (b) 2 (c) 3 (d) 1
39. 39.Which of these numbers is divisible by 4? (a) 1069 (b) 1067 (c) 1068 (d) 1075
40. 40.What is the remainder when 482 is divided by 9? (a) 6 (b) 4 (c) 7 (d) 5

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